Inequalities for the lattice width of lattice-free convex sets in the plane

TL;DR

This paper establishes sharp bounds relating the area and lattice width of planar lattice-free convex sets, characterizes those achieving the bounds, and connects these results to covering minima and optimization applications.

Contribution

It provides the first sharp upper bounds for area based on lattice width and characterizes the sets that attain these bounds, advancing understanding in lattice geometry.

Findings

Sharp upper bounds for area given lattice width

Characterization of sets attaining the bounds

Connections to covering minima and optimization applications

Abstract

A closed, convex set $K$ in $\mathbb{R}^2$ with non-empty interior is called lattice-free if the interior of $K$ is disjoint with $\mathbb{Z}^2$. In this paper we study the relation between the area and the lattice width of a planar lattice-free convex set in the general and centrally symmetric case. A correspondence between lattice width on the one hand and covering minima on the other, allows us to reformulate our results in terms of covering minima introduced by Kannan and Lov\'asz. We obtain a sharp upper bound for the area for any given value of the lattice width. The lattice-free convex sets satisfying the upper bound are characterized. Lower bounds are studied as well. Parts of our results are applied in a paper by the authors and Weismantel for cutting plane generation in mixed integer linear optimization, which was the original inducement for this paper. We further rectify a result of Kannan and Lov\'asz with a new proof.